Finite Group Databasebeta

← #475495

Subgroups — #475495

Order 512.

617 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11393C212yesmaximal
2#11389C212yesmaximal
3#11389C212yesmaximal
4#11320C212yesmaximal
5#11310C212yesmaximal
6#11309C212yesmaximal
7#10817C211yesmaximal, characteristic
8#10814C211yesmaximal, characteristic
9#10799C211yesmaximal, characteristic
10#3458C2212yes—
11C2×#405C2211yescharacteristic
12C2×#344C2212yes—
13C2×#344C2212yes—
14#1742C2212yes—
15#1740C2211yescharacteristic
16#1740C2212yes—
17#1740C2211yescharacteristic
18C2×#341C2211yescharacteristic
19#1735C2212yes—
20#1735C2211yescharacteristic
21#1733C2212yes—
22#1733C2212yes—
23#1733C2212yes—
24C2×#338C2211yescharacteristic
25C2×H(3,ℤ/4)C2212yes—
26C2×H(3,ℤ/4)C2212yes—
27C2×H(3,ℤ/4)C2212yes—
28C2×H(3,ℤ/4)C2211yescharacteristic
29C2×H(3,ℤ/4)C2212yes—
30#1729C2212yes—
31C22×M16C2312yes—
32C22×M16C2312yes—
33C22×M16C2311yescharacteristic
34C23×C8C2311yescharacteristic
35#544—24no—
36#544—42no—
37#544—42no—
38#543—42no—
39#533—24no—
40#533—24no—
41#533—42no—
42C2×#117C2312yes—
43C2×#117C2312yes—
44C2×#117C2312yes—
45C2×#117C2311yescharacteristic
46C2×#117C2312yes—
47#405—42no—
48C2×#97Q812yescomplemented
49C2×C4×C8Q812yescomplemented
50#344—24no—
51#344—42no—
52#343—42no—
53#342—41no—
54#342—42no—
55#342—41no—
56#341—41no—
57#340—41no—
58#340—42no—
59#339—24no—
60#339—24no—
61#339—42no—
62#338—41no—
63H(3,ℤ/4)—24no—
64H(3,ℤ/4)—24no—
65H(3,ℤ/4)—41no—
66H(3,ℤ/4)—42no—
67H(3,ℤ/4)—42no—
68#336—24no—
69C23×C4C2411yescharacteristic
70C2×M16—22no—
71C2×M16—21no—
72C2×M16—22no—
73C2×M16—21no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×Q811yescharacteristic
93C22×C8C2×Q811yescharacteristic
94C22×C8C2×Q812yes—
95C4⋊Q8—44no—
96C4⋊Q8—44no—
97C4⋊Q8—44no—
98#125—44no—
99#125—44no—
100#125—44no—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C4×Q8—48no—
105#117—22no—
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112#117—22no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—22no—
121#117#4114yes—
122#117#4114yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42C2×Q812yes—
137C2×C42#4114yes—
138#97—42no—
139C4×C8—42no—
140C24H(5,𝔽2)11yescharacteristic
141C2×Q8—48no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C22×Q811yescharacteristic
148C22×C4C2×#4112yes—
149C22×C4C2×#4112yes—
150C22×C4C2×#4112yes—
151C22×C4H(5,𝔽2)12yes—
152C22×C4H(5,𝔽2)11yescharacteristic
153C22×C4H(5,𝔽2)11yescharacteristic
154C22×C42−1+411yescharacteristic
155C22×C42−1+411yescharacteristic
156C22×C42−1+411yescharacteristic
157M16—42no—
158M16—42no—
159M16—42no—
160M16—42no—
161M16—42no—
162M16—42no—
163M16—42no—
164M16—42no—
165M16—42no—
166M16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—48no—
187C4⋊C4—48no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—22no—
206C42—24no—
207C42—22no—
208C42—24no—
209C42—24no—
210C42—24no—
211C42—42no—
212C42—42no—
213C42—42no—
214C42—42no—
215C23—81no—
216C23#53812yes—
217C23#54411yescharacteristic
218C23#55112yes—
219C23H(3,𝔽4)11yescharacteristic
220C23#56311yescharacteristic
221Q8—168no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—82no—
227C2×C4—84no—
228C2×C4—82no—
229C2×C4—162no—
230C2×C4—162no—
231C2×C4—162no—
232C2×C4#53311yescharacteristic
233C2×C4#53311yescharacteristic
234C2×C4#53311yescharacteristic
235C2×C4#53812yes—
236C2×C4#53812yes—
237C2×C4#54212yes—
238C2×C4#54212yes—
239C2×C4#54212yes—
240C2×C4#54311yescharacteristic
241C2×C4#54411yescharacteristic
242C2×C4#54411yescharacteristic
243C2×C4#55112yes—
244C2×C4#55112yes—
245C2×C4#55612yes—
246C2×C4#55612yes—
247C2×C4#55612yes—
248C2×C4#56311yescharacteristic
249C2×C4#56311yescharacteristic
250C8—41no—
251C8—41no—
252C8—42no—
253C8—82no—
254C8—82no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22#260511yescharacteristic
261C22#266511yescharacteristic
262C22#273612yes—
263C22#275912yes—
264C22#284411yescharacteristic
265C4—82no—
266C4—84no—
267C4—84no—
268C4—84no—
269C4—162no—
270C4#260511yescharacteristic
271C4#260511yescharacteristic
272C4#266511yescharacteristic
273C4#266511yescharacteristic
274C4#275912yes—
275C4#275912yes—
276C2—22no—
277C2—21no—
278C2—22no—
279C2—21no—
280C2—21no—
281C2#1705311yescharacteristic
subgroupG/H
centerC4#2605
commutator, Frattini, cosocleC23×C4C24
socleC2#17053
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
48. C2×#97Q8semidirect
49. C2×C4×C8Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#113932
#113892
#113202
#113102
#113092
#108172
#108142
#107992